P H Y S I C A L R E V I E W V O L U M E 1 4 0 , N U M B E R 4 B 22 N O V E M B E R 19 65

Non-Abelian Gauge Fields and Goldstone Bosons*

NORMAN FUCHS

Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania (Received 1 July 1965)

The theory of a massless, in general non-Abelian, gauge vector field is considered and the relation between the supplementary condition for the vector field and the condition for spontaneously broken symmetry is explored. Goldstone bosons are found to exist and their equations of motion are determined; the non-Abelian nature of the gauge group leads to interaction for the Goldstone bosons. The results are applied to the problems of broken isotopic and unitary symmetries.

I. INTRODUCTION

IN recent years, the possibility that a relativistic field theory, defined by a Lagrangian which is invariant

under a certain symmetry group, might possess un-symmetrical solutions has been discussed by a number of authors. It is generally believed that if this is the case, the "broken symmetry" solution must be de­generate in some sense and furthermore strongly interacting scalar bosons with zero mass must appear. This is the conjecture of Goldstone,1 for which Gold­stone, Salam, and Weinberg2 gave two proofs which were further generalized by Bludman and Klein.3

The proofs of existence of these massless boson excitations leave unanswered the question of how they interact. In the recent paper of Guralnik and Hagen,4

this problem is considered within the framework of a variety of field-theory models. They conclude that the zero-mass bosons may be either noninteracting gauge excitations, or that they had been put in the theory at the beginning, and hence the Goldstone theorem has no content. All symmetry groups that they consider are Abelian.

In this work we attempt to extend the results of Guralnik and Hagen to non-Abelian symmetry groups. Our frame of reference is the vector-meson picture of the strong interactions, as discussed, for example, by Sakurai5 and by Gell-Mann and Glashow.6 We find that owing to the non-Abelian nature of the gauge vector field, the Goldstone bosons may interact and thus give rise to physical effects. The broken symmetry we consider is of the "spontaneous type"; i.e., the true isotopic spin (unitary spin) currents are conserved, while the physical states are not invariant under the action of the symmetry group.

* Supported by the U. S. Atomic Energy Commission. 1 J. Goldstone, Nuovo Cimento 19, 154 (1961). 2 J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127,

965 (1962). 3 S. Bludman and A. Klein, Phys. Rev. 131, 2364 (1963). 4 G. S. Guralnik and C. R. Hagen, Imperial College, 1965 (un­

published report). 5 J. Sakurai, Ann. Phys. (N. Y.) 11, 1 (1960). 6 M. Gell-Mann and S. Glashow, Ann. Phys. (N. Y.) 15, 437

(1961).

II. FORMALISM

We consider the theory denned by the Lagrangian7

+lfWG,v- Gd^+^+^fr+Z (*), (1)

where the Lagrangian for the matter field <£ (\I/) is given by

£W = ̂ K+H^, (2) and the matter current fo1 is

ka»=&(x)a»TaxP(x). (3)

The group multiplication is defined by

i{<t>nh<t>v) — S fypattahcfyvc ,

i (0M/#„) = / < £ „ = — i<j>nlt(j)v ——% {<t>vt4>n) 3

a

(4)

and tabc are the (completely antisymmetric) structure constants of the Lie group. The Ta depend on the representation to which \// belongs. The Lagrangian is just that of Schwinger7 except for an additional %G2

term. The equations of motion

f2Gpiy= dyft>v— drf>p+i {^t^y), (5)

{dv-ilUt>v')G»v-d»G=fr, (6)

d^=G, (7)

[ ^ ( d M - f T ^ ) + / 3 w > = 0 , (8)

are the same except for Eq. (7) where Schwinger has dM<£"=0. The canonical commutation relations are the same, namely,

tC**a(a?) ,Gfc W (« / ) ]=«a6«* I f i (x -X / ) ,

i[<t>a°(x),Gb(x')l = 5ab8(x-x'), (9)

{\!sa(x)rtp(x')} = 5ap5(x-x'),

and all other commutators vanish. The current &

' J. Schwinger, Phys. Rev. 125, 1043 (1962); 130, 402 (1963).

B911

B912 NORMAN F U C H S

satisfies the "conservation" law

(d M - i c % , )* M =0 , (10)

and G, the canonical momentum conjugate to <£o, obeys

(d M -* '% , )d M G=0. (11)

If we define the current j ^ by

j^dfi+iG'tyJ, (12)

then jn satisfies a true conservation law

drf»=0, (13)

and so we may construct the generator of constant infinitesimal gauge transformations

' - - / •

dsx(d0G+iG't<l>o')S\

(14)

= — / jo dzxd\.

The generator Gs\ is time-independent, and induces the isotopic-spin transformations (with constant SX)

GpV-*(l+i't8k')GpP,

<^-> ( 1 + ^ 5 X 0 ^ ,

G-*(l+i'td\>)G,

*->(i+*'m')*, (15)

More generally, if we do not demand that 8X be a constant, the generator of gauge transformations is then

' - / •

dzx[Gd05\-8\(doG+iG't<l>o')l. (16)

The transformations induced by this operator are

G„„-> (l+i'tdX^G^,

G-^(l+i't5\>)G,

f-*(l+i'T8k')f, (17)

<!>»-> (l+i'tdX'fa+duffk,

As Schwinger points out, one may not now conclude that £ would be invariant or Gg\ time-independent if 8X satisfies

( d / , - i ' % , ) ^ X = 0 , (18)

since this implies that 5X must be an operator, not a numerical gauge function. This is an essential difference between Abelian and non-Abelian gauge theories. In the Abelian gauge limit, the above reduces to the usual formalism.8

8 J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1959), p. 84.

The energy-momentum operators 60k, 600 are given by

r f c °= / 2 G^G^-^ (a & - iT^ ,o^+^^z^ ,

r°°=i/2C(G°*)2+i(Gw)2]

and are identical to those of Schwinger except for the — |G 2 term in 000. The commutation relations for these operators satisfy the conditions

- i [ ^ (#),0oo(x')] = - (0°*(x)+dok(x'))dkd(x-x'),

- 4 T ° H x ) , r 0 0 (*')]== - (Tok(x)+Tok(x'))dk8(x-x'). (20)

Note that the new energy-density 000 is gauge-invariant for constant gauge function 8X. However, if we write 000

in the form

0oO== r oo_iG2+[G 2 +^a o G--Gao0 0 ] (21)

and notice that the bracketed expression is stable under variations in G, we see that the energy spectrum is not bounded below. We will comment on this problem below.

III. BROKEN AND UNBROKEN SYMMETRIES

If we demand that for physical states | \£)

G | * ) = 0 = d 0 G | * > , (22)

then G5\|^r)==0 and the unitary operator Us\, denned by

£/sx=exp(iG8X), (23)

will leave all physical states invariant. This is analogous to the usual supplementary condi­

tion of quantum electrodynamics and leads to a sym­metric solution. However, if we do not impose this, then we will obtain a theory with broken symmetry. In the case of an Abelian gauge, we may define the de­generate vacua by

|5X)=*73x|0> (24)

for all 8X satisfying

D5X-0 . (25)

In particular, if we choose

Vii— dfl8\= constant, (26)

then we recover the results of Guralnik and Hagen, and the gauge excitations G are indeed nonobservable since G satisfies a free-field equation. In the non-Abelian-gauge case, however, as we emphasized above, one may allow only constant gauge functions 8\ in Eq. (24). Since G in this case does not necessarily satisfy a free-field equation, it may not necessarily be considered a gauge excitation without physical effects.

If now <j> {x) is a scalar field (in general, consisting of

N O N - A B E L I A N GAUGE F I E L D S AND G O L D S T O N E BOSONS B 913

a product of field operators) then

<0|[G«x,*(*)]|0>=^,<0|*(*)|0> = 0 if GsxiOHO. (27)

Hence we see that the supplementary condition Eq. (22) is intimately related to the Goldstone theorem, in the sense that the Goldstone theorem applies if and only if |0) is not a state satisfying Eq. (22).

Now, for example, consider SU(2) and a scalar isotriplet <£= (<£i,02,03); broken symmetry is defined by

<0|*,|0>*0,

<0ki,2|0>=0. KZ*}

Putting this into Eq. (22) we see that at least one of Gi,2 must satisfy

Gil 0>^0, (29)

and so G»- excites a state | Si) from the vacuum which is degenerate in energy with the vacuum. Thus the Goldstone theorem holds. Now, however, the field operator d satisfies the equation of motion

nGi^dpTf, (30)

where P* is the usual isotopic spin current

T^kP-H&rtGP"), (31)

and Tf is not conserved, since | Si) has charge and its contribution to the total isotopic spin current is not included in TV*. We should like to point out that the Goldstone bosons exist and interact only by virtue of the symmetry breaking, i.e., the strength of their inter­action vanishes in the limit of no symmetry breaking.9'10

On the other hand, we see that in the non-Abelian case, if the Goldstone bosons do exist then they must interact in order to maintain the broken symmetry. Since the Goldstone bosons G»- arising from broken isotopic spin conservation must be charged, the isotopic spin current T** of the system excluding Gi cannot be conserved, and from Eq. (30) we see that this implies interaction. Furthermore, it is clear that the "incom­plete" nature of the multiplet of Goldstone bosons is not in that only part of a multiplet exists, but is mani­fested by some members interacting and others being free (to a first approximation). In the case of SU(3),

9 Y. Nambu and J. Sakurai, Phys. Rev. Letters 11, 42 (1963). 10 D. Horn, Nuovo Cimento 29, 571 (1963).

broken symmetry requires the existence of a set of n mesons coupled to the divergence of the strangeness-changing currents of the other fields; this is the result of Nambu and Sakurai.9 However, the remaining members of the octet are coupled as well to the diver­gences of the other currents, which in lowest approxi­mation vanish.

In the case of a spontaneous breakdown of SU(3) symmetry, as described by Freund and Nambu,11 "the baryon field operators have exact octet transformation properties under SU(3), but the vacuum state now transforms according to a continuum dimensional representation of SU(3)." The extra terms induced by the symmetry breaking in the matrix elements of the conserved currents 7M are easily seen to arise from the term dMG in the expression for /M obtained from Eqs. (31) and (6):

J^T^-dfi, (32)

where we recall that TM is the current of the fields other than G. The results of Freund and Nambu then follow.

There are at least two unresolved problems in the theory developed above. One is the lack of a lower bound to the energy spectrum. This might be remedied by adding a XG4 meson-meson scattering term to the Lagrangian, but it is difficult to see how this could be done without doing violence to the theory as presented. Alternatively, this problem might be remedied in a Gupta-Bleuler fashion by imposing the weaker condi­tion G ( + ) | ^ )=0 on physical states |\F). However, this condition relates the scalar mesons G to the longitudinal photons <£o, and we want these fields to be independent. This brings us to the second problem: zero-mass particles. The vector gauge particles as well as the Goldstone bosons have vanishing bare mass. Various authors12 have proposed arguments in favor of the possibility of massive vector fields with zero bare mass; however, even accepting this still leaves open the question of the renormalized mass of the Goldstone bosons.

ACKNOWLEDGMENT

The author wishes to thank Professor S. Bludman for his interest and for valuable discussions.

11 P. G. O. Freund and Y. Nambu, Phys. Rev. Letters 13, 221 (1964).

12 J. Schwinger, Phys. Rev. 125, 397 (1962).